Residually finite actions and crossed products

TL;DR

This paper explores the concept of residual finiteness in group actions on compact spaces and establishes a connection to the algebraic property of having norm microstates in the associated crossed product, specifically for free groups.

Contribution

It proves that residual finiteness of free group actions on zero-dimensional spaces is equivalent to their crossed product being an MF algebra with norm microstates.

Findings

Residual finiteness characterizes free group actions on zero-dimensional spaces.

Crossed products of such actions are MF algebras if and only if the actions are residually finite.

Abstract

We study a notion of residual finiteness for continuous actions of discrete groups on compact Hausdorff spaces and how it relates to the existence of norm microstates for the reduced crossed product. Our main result asserts that an action of a free group on a zero-dimensional compact metrizable space is residually finite if and only if its reduced crossed product admits norm microstates, i.e., is an MF algebra.